Modeling and experimental circuit implementation of fractional single-transistor chaotic oscillators

Abstract

This study presents the first experimental realization of a single-transistor fractional chaotic oscillator, obtained by extending a minimalistic integer-order circuit by systematically transforming its reactive components, namely two inductors and a capacitor, into fractional elements. Starting from the Grünwald-Letnikov definition and using a string structure finite-order approximation for implementation, the dynamics are studied over a range of fractional orders. Results from equation models, circuit simulations, and experimental measurements are juxtaposed, yielding broadly consistent results. The introduction of fractional elements is found to have profound effects on the chaotic dynamics, influencing oscillation amplitude, spectral flatness, and bifurcation characteristics. In particular, inspection of the resulting Poincaré sections reveals a gradual distortion of the interplay between the relaxation and resonance aspects of the circuit dynamics with decreasing fractional order. While less generative than other manipulations, such as inserting fractal resonators, the ability to introduce fractional components into elementary nonlinear oscillator circuits provides a new, highly versatile, and compact physical tool. Potential applications include modeling electronically real-world phenomena endowed with considerable memory and nonlocality, such as neural activity and viscoelasticity.